Singular value decomposition (svd) in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of SVD on a simple matrix

What is the output of the following MATLAB code snippet?

MATLAB

A = [3 1; 1 3];
[U,S,V] = svd(A);
disp(diag(S)');

A [2 4]

B [4 2]

C [5 1]

D [3 3]

💡 Hint

Recall that singular values are the square roots of eigenvalues of A'*A.

The matrix A'*A has eigenvalues 16 and 4, so singular values are sqrt(16)=4 and sqrt(4)=2.

🧠 Conceptual

intermediate

Properties of matrices in SVD

Which statement about the matrices U, S, and V from the SVD of a matrix A is correct?

A U and V are orthogonal matrices, and S is a diagonal matrix with non-negative entries.

B U and V are diagonal matrices, and S is an orthogonal matrix.

C U is upper triangular, V is lower triangular, and S is symmetric.

D U and V are symmetric matrices, and S is a diagonal matrix with negative entries.

💡 Hint

Think about the definition of orthogonal matrices and singular values.

In SVD, U and V are orthogonal matrices (U'*U=I, V'*V=I), and S is diagonal with non-negative singular values.

🔧 Debug

advanced

Identify the error in SVD reconstruction

What is the error in the following MATLAB code that attempts to reconstruct matrix A from its SVD?

MATLAB

A = [1 2; 3 4];
[U,S,V] = svd(A);
A_reconstructed = U * S * V;
disp(A_reconstructed);

A There is no error; the code reconstructs A correctly.

B The svd function returns V transposed, so V should be transposed again.

C The multiplication order is incorrect; it should be U * S * V'.

D The matrix S is not diagonal, so reconstruction fails.

💡 Hint

Recall the formula for reconstructing A from U, S, and V.

The correct reconstruction is A = U * S * V', not U * S * V.

📝 Syntax

advanced

Syntax error in SVD usage

Which option contains a syntax error when performing SVD in MATLAB?

A [U,S,V] = svd A;

B [U,S,V] = svd(A);

C [U,S,V] = svd(A, 'econ');

D [U,S,V] = svd(A,0);

💡 Hint

Check the correct syntax for function calls in MATLAB.

In MATLAB, function arguments must be enclosed in parentheses; 'svd A;' is invalid syntax.

🚀 Application

expert

Number of singular values greater than a threshold

Given matrix A, how many singular values are greater than 1.5 in the following code?

MATLAB

A = [4 0 0; 0 3 0; 0 0 0.5];
[~,S,~] = svd(A);
sv = diag(S);
count = sum(sv > 1.5);

A 0

B 3

C 1

D 2

💡 Hint

Look at the diagonal entries of S and count those greater than 1.5.

The singular values are 4, 3, and 0.5; only 4 and 3 are greater than 1.5, so count is 2.